A323940 Number of nonisomorphic systems (isomers) for the unsymmetrical schemes (group C_s) for unbranched tri-4-catafusenes as a function of the number of hexagons (see Cyvin et al. (1996) for precise definition).
0, 1, 8, 52, 244, 1093, 4490, 17952, 69304, 262385, 973916, 3562532, 12856716, 45880933, 162085694, 567578784, 1971766704, 6801381633, 23309759728, 79421199860, 269160256356, 907726205221, 3047449152562, 10188384019072, 33930769372904
Offset: 3
Links
- Georg Fischer, Table of n, a(n) for n = 3..103
- S. J. Cyvin, B. N. Cyvin and J. Brunvoll, Isomer enumeration of some polygonal systems representing polycyclic conjugated hydrocarbons, Journal of Molecular Structure 376 (Issues 1-3) (1996), 495-505. See Table 1 on p. 500.
- Eric Weisstein's World of Mathematics, Fusene.
- Wikipedia, Molecular symmetry.
- Wikipedia, Point groups in three dimensions.
- Wikipedia, Polyhex (mathematics).
- Wikipedia, Schoenflies notation.
- Index entries for linear recurrences with constant coefficients, signature (14,-71,116,259,-1246,1013,2520,-5187,594,5931,-4428, -1215,2430,-729).
Programs
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Maple
# Calculates a(r) = AA(r), where r = n is the number of hexagons. # Crude numbers: JJ := proc(i) sum(binomial(j + 1, 3)*binomial(i - 2, j - 1)*2^(i - 1 - j), j = 1 .. i - 1); end proc; # Linearly annelated systems of D_{2h} symmetry: DD := proc(r) 1/4*(1 - (-1)^r)*(r - 1); end proc; # Linearly annelated systems of C_{2v} symmetry: LL := proc(r) 1/2*binomial(r, 3) - (1/8 - 1/8*(-1)^r)*(r - 1); end proc; # Centrosymmetrical (C_{2h}) systems: CC := proc(n) 1/24*(1 - (-1)^n)*((3 + n)*3^(1/2*n - 3/2) - 3*n + 3); end proc; # Total mirror-symmetrical (C_{2v}) systems: MM := proc(n) CC(n) + LL(n); end proc; # Unsymmetrical (C_s) systems: AA := proc(r) 1/4*(JJ(r) - DD(r) - 2*CC(r) - 2*MM(r)); end proc; # Generate sequence: for m from 3 to 100 do AA(m); end do; # Petros Hadjicostas, May 26 2019 -
Mathematica
LinearRecurrence[{14, -71, 116, 259, -1246, 1013, 2520, -5187, 594, 5931, -4428, -1215, 2430, -729}, {0, 1, 8, 52, 244, 1093, 4490, 17952, 69304, 262385, 973916, 3562532, 12856716, 45880933}, 100] (* from the g.f., Georg Fischer, Nov 07 2019 *)
Formula
a(n) = (1/8) * (1 - (-1)^n) * ((n - 1) - (n + 3) * 3^((n - 5)/2)) + (1/8) * (n^2 + 11 * n + 12) * (n - 2) * 3^(n - 6) - (1/4) * binomial(n, 3) for n >= 3. - Petros Hadjicostas, May 26 2019
G.f.: x^4*(1 -6*x +11*x^2 -32*x^3 +182*x^4 -346*x^5 -122*x^6 +950*x^7 -831*x^8 +336*x^9 -297*x^10 +90*x^11) / ( (1+x)^2*(3*x^2-1)^2*(3*x-1)^4*(x-1)^4 ). - R. J. Mathar, Jul 25 2019
Extensions
Name edited by Petros Hadjicostas, May 26 2019
More terms using various equations in Cyvin et al. (1996) from Petros Hadjicostas, May 26 2019
Comments